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where Vd is volume of distribution, and Kis the first-order kinetic rate constant of elimination. According to Eq. (1.7) the linear relationship between dose and Cp holds for all sized doses.

If for the same one-compartment model the input is changed from an intravenous bolus to first-order kinetic input (e.g., gut absorption), the expected Cp versus time curves are shown in Fig. 1.14. The kinetics for this system are described by d(Cp )

p where ka is the first-order kinetic rate input constant, and Ca is the driving force concentration or concentration of drug at the site of administration. The integrated solution for Eq. (1.9) is given by Eq. (1.10):

Although Eq. (1.10) is linear with respect to dose, it is not linear with respect to its parameters (ka and K). The definition of linear and nonlinear pharmacokinetic models is based on the relationship between Cp and dose, not with respect to the parameters.

Nonlinear pharmacokinetics. Nonlinear pharmacokinetics simply means that the relationship between dose and Cp is not directly proportional for all doses. In nonlinear pharmacokinetics, drug concentration does not scale in direct proportion to dose (also known as dose-dependent kinetics). One classic drug example of nonlinear pharmacokinetics is the anticonvulsant drug phenytoin.38 Clinicians have learned to dose pheny-toin carefully in amounts greater than 300 mg/day; above this point, most individuals will have dramatically increased phenytoin plasma levels in response to small changes in the input dose.

Many time-dependent processes appear to be nonlinear, yet when the drug concentration is measured carefully relative to the time of dose, the underlying dose-to-drug-concentration relationship is directly proportional to the dose and therefore is linear (see "Time- and State-Varying Pharmacokinetics and Pharmacodynamics").

1.4.2 Time- and state-varying pharmacokinetics and pharmacodynamics

Time- and state-varying pharmacokinetics or pharmacodynamics refer to the dynamic or static behavior of the parameters used in the model. Time-varying would encompass phenomena such as the circadian variation of Cp owing to underlying circadian changes in systemic clearance. While time-varying can be considered a subset of the more general state-varying models, state-varying parameters can change as an explicit function of time and/or as an explicit function of another pharmacokinetic or pharmacodynamic state variable (e.g., metabolite concentration, AUC, etc.).

Time-varying. Figure 1.3 shows two possible Cp versus time plots that could arise from a pharmacokinetic/pharmacodynamic system where Cl (bottom panel) or receptor density (top panel) varies sinusoidally